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A Numerical Calculation Method of Apparent Contact Angles on Heterogenous Double Roughness Surfaces Jian Dong, Yanli Jin, He Dong, and Li Sun Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02564 • Publication Date (Web): 08 Sep 2017 Downloaded from http://pubs.acs.org on September 11, 2017
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A model roughness geometry for theoretical analysis. (a) The fine scale roughness—the first generation. (b) The fine scale roughness forms the surface of the coarse scale pillars—the second generation of roughness. (c) The pillar geometry at both scales is assumed periodic. The top view of one period is shown. (From Ref 32. By Professor NA. Patankar). 78x84mm (96 x 96 DPI)
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System model composed of a droplet and a regular double-roughness surface. (a) Three-dimensional system model. (b) Profile of system model. (c) Sketch of the first and second generation structures. 135x115mm (96 x 96 DPI)
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Four kinds of wetting contact states of a droplet on a double-roughness surface. (a) W-W state. (b) C-W state. (c) W-C state. (d) C-C state. 78x90mm (96 x 96 DPI)
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ACAs and wetting contact states are the function of the relative structure height h1/a1 and h2/a2 (a1=b1=100nm, a2=b2=10µm). The materials of the first generation structure, the second generation structures and the base are as follows: (a) Si, PDMS and Si; (b) Si, PC and Si; (c) SiO2, PDMS and Si; (d) SiO2, PDMS and SiO2. 141x102mm (96 x 96 DPI)
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SEM images of double-roughness surfaces taken from literatures (a) The lotus leaf surface with dual-scale structure and a thin wax film. (b) The heterogenous double-roughness surface after 10s silver deposition on copper substrate. (c) The heterogenous double-roughness surface after 30s silver deposition on copper substrate. ((a) From Ref 48 and (b) (c) from Ref 49). 195x50mm (96 x 96 DPI)
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A Numerical Calculation Method of Apparent Contact Angles on Heterogenous Double Roughness Surfaces 1
Jian Dong*, Yanli Jin, He Dong and Li Sun
2
Key Laboratory of E&M, Zhejiang University of Technology, Ministry of Education &
3
Zhejiang Province, Hangzhou, 310014, China
4
ABSTRACT: Double-roughness surfaces can be used to mimic lotus surfaces. The
5
apparent contact angles (ACAs) of droplets on these surfaces were first calculated by
6
Herminghaus. Then Patankar utilized pillar model to improve the Herminghaus approach
7
and put forward the formulae for ACAs calculation of the homogeneous double-roughness
8
surfaces where the dual-scale structures and the bases were the same wettable materials. In
9
this paper, we propose a numerical calculation method of ACAs on the heterogenous
10
double-roughness surfaces where the dual-scale structures and the bases are made of
11
different wettable materials. This numerical calculation method has successfully enhanced
12
the Herminghaus approach. It is promising to become a novel design approach of
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heterogenous superhydrophobic surfaces, which are frequently applied in technical fields
14
of self-cleaning, anti-icing, anti-fogging and enhancing condensation heat transfer.
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KEY WORDS: double-roughness surfaces, apparent contact angles, a numerical
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calculation method, different wettable materials.
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1. INTRODUION
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In recent years, the research on the wettability of the solid surfaces has attracted great
6
attention due to their special properties1–9. SHSs (surfaces on which ACAs larger than
7
150°) have many important applications such as droplet manipulation10, self-cleaning11–13,
8
anti-icing14-16
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superhydrophobicity because of double structured roughness. Researchers have studied
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many double-roughness surfaces to gain artificial SHSs19–25 imitating lotus leaves. The
11
ACA value on SHSs is an important criterion to evaluate the superhydrophobicity of SHSs.
12
Thus, it is of great significance to calculate the ACAs on double-roughness surfaces.
and
protein
adsorption17,18.
The
lotus
leaves
have
excellent
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There have been some studies about wetting behavior of rough surface. The earliest
14
theoretical work was attributed to Wenzel26 and Cassie27 whose models respectively
15
predict the wetted contact angle and the composite contact angle of a drop on a rough
16
surface. Herminghaus28 first proposed the general approach to predict the composite
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contact angle on hierarchical surfaces by studying the wetting properties of plant leaves.
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Based on energy considerations, the ACA is given by
19
cos θ n +1 = (1 − wn ) cos θ n − wn
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where n is the generation number of the indentation hierarchy, wn is the area fraction
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of the free liquid surfaces suspended on n generation, larger n corresponds to larger 2 ACS Paragon Plus Environment
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length scale, θ n is the ACA on n generation, and θ 0 is the equilibrium contact angle
2
of the liquid drop on the flat surface. Bormashenko et al.29,30 did experiments and verified
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the formulae about the ACAs of droplets on double-roughness surfaces. Patankar31,32
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further classified the droplet wetting states on homogeneous double-roughness surfaces
5
and enhanced the Herminghaus approach using pillar model. As is shown in Figure 1,
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Patankar assumed that the first generation structure (nano structure), the second generation
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structure (micro structure) and the base are made of the same wettable material. Square
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pillars are arranged in a regular array framework and form the hierarchy surface. At the
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first (second) generation, a（ , b（ and H（ 1 a2） 1 H 2） are the side length of square pillar, 1 b2）
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the periodic spacing of regular array and the height of square pillar, respectively.
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According to the pillar model, Patankar got the formulae of ACA θ r as follows:
12
Cassie state:
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15 16 17
18
cosθ rc =A1 (1 + cosθ 0 ) − 1
(2)
4 A1 w cosθ r = 1 + cosθ 0 a 1 H1
(3)
Wenzel state:
where A1 =
1 b1 + 1 a1
2
Patankar firstly judged the wetting contact state of the first generation structure (Cassie or
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Wenzel) and used eq 2 or 3 to get θ1 (the ACA of the droplet on the first generation).
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Then, once again he judged the wetting contact state of the second generation structure
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(Cassie or Wenzel). Finally, he calculated out the ACA of the droplet on the
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double-roughness surface using the same formula as eq 2 or 3 with subscripts “1” replaced
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by “2” and θ 0 replaced by θ1 .
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Figure 1. A model roughness geometry for theoretical analysis. (a) The fine scale
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roughness—the first generation. (b) The fine scale roughness forms the surface of the
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coarse scale pillars—the second generation of roughness. (c) The pillar geometry at both
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scales is assumed periodic. The top view of one period is shown. (From Ref 32. By
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Professor NA. Patankar).
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After these works, Bhushan et al.33 further proposed the importance of low contact
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angle hysteresis for droplet-easy-rolling property. Starov et al.34 discussed the influence of
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the long-range surface forces on multi-scale partially wetted surfaces and contact angle 4 ACS Paragon Plus Environment
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hysteresis. Liu et al.35 have declared that the micro-nano hierarchical roughness is the key
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structural factor for keeping condensed droplets in Cassie state on SHSs. Sajadinia et al.36
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predicted the ACAs of the dual-scale rough surfaces with smooth sidewalls and found the
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interaction between microscale and nanoscale roughness on wettability of surfaces. Bell, et
5
al.37 proposed that the nanoscale roughness plays an important role in increasing effective
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Young's angles of the microscale features. Wu et al.38 contrasted the ACAs of
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double-roughness surfaces with structured sidewalls and with smooth sidewalls. However,
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all the methods, except the Herminghaus model, are only suitable for homogeneous
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double-roughness surfaces composed of the same wettable materials. In fact, surfaces are
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always decorated by different wettable dual-scale structures. For example, hydrophobic
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bases and hydrophobic microstructures are always decorated by many hydrophilic
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nano-pillars to enhance dropwise condensation nucleation. All their methods can not
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calculate the ACAs on heterogenous double-roughness surfaces.
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In this paper, we propose a numerical calculation method to calculate the ACAs on
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heterogenous double-roughness surfaces composed of different wettable materials. The
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thermodynamic analysis and the numerical calculation method presented in this paper can
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be used as a new design approach for heterogenous double-roughness SHSs.
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2. THEORETICAL MODEL
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2.1 Parameters and Wetting Contact States
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For the sake of simplicity, we consider a system made up of a droplet and a regular 5 ACS Paragon Plus Environment
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double-roughness surface on which the first and second generation structures are both
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square pillars and the geometry scales are periodic. Particularly, the dual-scale structures
3
and the base are made of different wettable materials shown in Figure 2.
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When the drop radius is less than the critical value of
rcap
(where
ργ cap = γ LV / ρ
,
γ LV
5
is the liquid surface tension, ρ the liquid density and g the gravity acceleration, 2.7
6
mm for water drop), the effects of the gravity, the line tension and the curvature of the
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liquid-vapor interface in the groove can be neglected. For the small droplets (drop radius
8
r < rcap
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(1) The shape of the droplet is coronal, and the contact line between the droplet and the
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), the following assumptions are valid36,39：
base is a circle. (2) The curvature of the liquid-vapor interface in the groove and the volume of the droplet inside the gap can be neglected.
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(3) The influence of the gravity and the line tension can be neglected.
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(4) The equilibrium shape of the droplet is finally stable in the minimum free energy
15
state.
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(b)
(a)
6 7 8 9
(c)
10 11 12 13 14 15
Figure 2. System model composed of a droplet and a regular double-roughness surface. (a)
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Three-dimensional system model. (b) Profile of system model. (c) Sketch of the first and
17
second generation structures.
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The system is made up of a droplet and a square area whose sidelength L0 is constant
19
and greater than the droplet diameter. The scale of the first generation structure is far less
20
（ than the second generation structure. a（ , b（ , x（ , Sext and S base 1 a2） 1 b2）, h 1 h2） 1 x2）
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are the side length of square pillar, the periodic spacing of regular array, the height of
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square pillar, the penetration depth of droplet in the first (second) generation structure, the
23
external surface area and the apparent contact area, respectively. Base on aforementioned 7 ACS Paragon Plus Environment
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assumptions, the total volume of the droplet V is considered to be the spherical volume,
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f1 ( f 2 ) is the solid fraction of the first (second) generation structure which is ratio of
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solid areas in the apparent contact surface to apparent areas, and r1 ( r2 ) is the roughness
4
factor of the first (second) generation structure which is ratio of total actual areas to
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apparent areas (The relationships between parameters can be seen in S1 in Supporting
6
Information).
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As is shown in Figure 3, the droplet on double-roughness surface has four different kinds of wetting contact states40: (a) The first and second generation structures are both in Wenzel state, that is, x1 = h1
10
and x2 = h2 , namely W-W state.
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(b) The first generation structure is in Cassie state or the contact state transiting from Cassie to Wenzel, and the second generation structure is in Wenzel state, that is,
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0 ≤ x1 < h1 and x2 = h2 , namely C-W state.
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(c) The first generation structure is in Wenzel state, the second generation structure is in
15
Cassie state or the contact state transiting from Cassie to Wenzel, that is, x1 = h1 and
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0 ≤ x2 < h2 namely W-C state.
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(d) The first and second generation structures are both in Cassie state or the contact state transiting from Cassie to Wenzel, that is, 0 ≤ x1 < h1 and 0 ≤ x2 < h2 , namely C-C state.
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(a)
(b)
(c)
(d)
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Figure 3. Four kinds of wetting contact states of a droplet on a double-roughness surface.
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(a) W-W state. (b) C-W state. (c) W-C state. (d) C-C state.
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2.2 Dimensionless Free Energy and ACA Numerical Calculation
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The free energy includes the interfacial free energy, the potential energy and the energy of
16
line tension. In the model, the influence of the gravity and the line tension can be
17
negligible. The free energy of system G ( x1 , x2 , θ ) is approximately equal to the
18
interfacial free energy and expressed by41–43:
G ( x1 , x2 , θ ) =γ LV SLV +γ SL0 SSL0 +γ SL1SSL1 +γ SL2 SSL2 19
+γ SV0 SSV0 +γ SV1SSV1 +γ SV2 SSV2
(4)
20
where γ and S are the interfacial tension and the area of the interface, respectively;
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subscripts LV , SL and SV denote the liquid-vapor, solid-liquid and solid-vapor
22
interface, respectively; subscripts 1, 2 and 0 denote the first generation structure, the
23
second generation structure and the base, respectively. 9 ACS Paragon Plus Environment
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According to four different kinds of wetting contact states, we can obtain the
2
2， 3， 4 ) (see part S2 in Supporting corresponding free energy of system GI ( x1 , x2 , θ )( I = 1，
3
Information). When the double-roughness surface and the total volume of the droplet are
4
2， 3， 4 ) can be defined, the corresponding dimensionless free energy G 'I ( x1 , x2 , θ )( I = 1，
5
also deduced (see part S2 in Supporting Information).
6
W-W State: 2
1 3 = G '1 ( x1 , x2 , θ ) 2 − 2 cos θ − sin 2θ (1 − f1 )(1 − 3 2 − 3cos θ + cos θ − f 2 ) cos θ e 0 + r2 ( r1 + f1 − 1) cos θ e1
}
+ (1 − f1 )( r2 + f 2 − 1) cos θ e2
7 8
{
(5)
C-W State: 2 3
G '2 ( x1 , x2 , θ ) r sin 2θ (1 − f1 ) + 2 − 2 cos θ = 3 2 2 − 3cos θ + cos θ 1
4x − r2 f1sin 2θ 1 + 1 cos θ e1 a1
9 10
(6)
W-C State: 2
1 3 = G '3 ( x1 , x2 , θ ) 2 − 2 cos θ + sin 2θ (1 − f 2 ) 3 2 − 3cos θ + cos θ 4 x − sin 2θ ( r1 + f1 − 1) cos θ e1 + (1 − f1 ) cos θ e2 1 + 2 f 2 a2
11 12
C-C State:
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2
1 3 = G '4 ( x1 , x2 , θ ) 2 − 2 cos θ + sin 2θ 1 − f 2 3 2 − 3cos θ + cos θ 4 x 4 x + f 2 (1 − f1 ) 1 + 2 − sin 2θ f1 f 2 1 + 1 1 a2 a1 + 1
4 x2 cos θ e1 a2
(8)
2
where θ e1 , θ e2 and θ e0 are the equilibrium contact angle on the flat surfaces referring
3
to the first generation structure, the second generation structure and the base, respectively.
4
A droplet on a surface with its initial wetting contact state has a relatively high free
5
energy . Then the droplet changes its shape to reach the final equilibrium state with
6
minimum free energy, which is certainly among the four wetting contact states. When the
7
system is in the minimum dimensionless free energy, θ is the ACA and the
8
corresponding wetting contact state is the final wetting contact state. According to this
9
principle, we design an ACA numerical calculation method (see part S3 which includes
10
Scheme S1 and a corresponding C++ program). Firstly，we set all the possibilities through
11
the change of parameters x1 (0 ≤ x1 ≤ h1 ) , x2 (0 ≤ x2 ≤ h2 ) and θ (0 < θ < 180 ) to
12
calculate out all possible G 'I (I = 1, 2,3, 4) . Subsequently, we find out the partial minimum
13
G 'I min min = G 'I (I 1, 2,3, 4) and dimensionless free energy in four contact states using=
14
record the corresponding θ I (I = 1, 2,3, 4) . Then, we find out the whole minimum
15
= G 'min min = G 'I min (I 1, 2,3, 4) dimensionless free energy using
16
corresponding θ I as the ACA. Also, the subscript I denotes the final droplet wetting
17
contact state. 11 ACS Paragon Plus Environment
and record the
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3. RESULTS AND DISCUSSION
2
3.1 ACA Numerical Calculation
3
3.1.1 Dual-scale Structure and Base Made of the Same Material
4
When the dual-scale structure and the base are all made of the same material, the
5
numerical calculation method is equal to the Patankar approach. Assuming that the
6
dual-scale structure and the base are all made of silicon (Si) and the liquid is water, we
7
conclude that θ e1 =θ e2 =θ e0 =98° 44. We select different dimensions of the dual-scale
8
structure and respectively calculate out ACAs using the numerical calculation method and
9
the Patankar approach shown in Table 1. We can get θ rN = θ rP . That is, when the
10
dual-scale structure and the base are the same material, the numerical calculation method
11
in this paper should be all in agreement with the Patankar approach.
12
Table 1. ACA results from the numerical calculation method and the traditional approach
13
concerning the surfaces made of the same materiala a1 ( nm ) b（ 1 nm）
100 200 100 100 100 100 100 14
a
100 100 200 100 100 100 100
h（ 1 nm）
200 200 200 100 200 200 200
a2 ( μm ) b2 ( μm )
10 10 10 10 20 10 10
10 10 10 10 10 20 10
h2 ( μm )
20 20 20 20 20 20 10
Results from the numerical calculation State θ rN (°) W-C W-C W-W W-C W-C W-W W-W
148.7 147.9 142.1 145.1 137.8 142.1 146.6
Results from the Patankar approach State W-C W-C W-W W-C W-C W-W W-W
θ rP (°) 148.7 147.9 142.1 145.1 137.8 142.1 146.6
The liquid is water , the material Si, θ rN results from the numerical calculation and θ rP
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results from the Patankar approach.
2
3.1.2 Dual-scale Structure and Base Made of Different Materials
3
When the dual-scale structure and the base are made of different materials, the Patankar
4
approach can not be used to deal with this circumstance. The Herminghaus model can be
5
used to calculate out ACAs on heterogenous hierarchical surfaces only when droplets are
6
in C-C wetting contact state. We propose a numerical calculation method to calculate out
7
ACAs on heterogenous double-roughness surfaces. We select four different wettable
8
materials, which are Si (hydrophobicity), PDMS (hydrophobicity)45, Polycarbonate (PC,
9
hydrophilicity)46 and silicon dioxide (SiO2, hydrophilicity)47 , to form four different
10
“virtual” heterogenous hierarchical surfaces. We use the numerical calculation method to
11
calculate out corresponding ACAs and wetting contact states of droplets on these surfaces
12
shown in Table 2. The ACA for Type 1 in Table 2 (164.3°) can be also derived from eq 1
13
in the Herminghaus model. The Herminghaus model seems to be accurate and valid.
14
Above all, the numerical calculation method in this paper has successfully improved the
15
traditional ACA calculation theory.
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Table 2. ACA numerical calculation results for surfaces made of different materialsa
1
Types
Generation
Material
θ e (°)
1
First Second Base First Second Base First Second Base First Second Base
PDMS Si Si PC Si Si PDMS SiO2 Si PDMS SiO2 SiO2
113.7 98 98 81.1 98 98 113.7 65.3 98 113.7 65.3 65.3
2
3
4
State
θ rN (°)
C-C
164.3
W-W
42.8
W-C
154.0
W-C
154.0
a= b= h1 / 2= 100nm , a2= b2= h2 / 2= 10μm . 1 1
2
a
3
3.2 Influence of the Relative Structure Spacing on ACAs
4
Figure 4 gives the influence of the relative spacing on ACAs and wetting contact states in
5
four different “virtual” surfaces. The C-C state has a small contact angle hysteresis and is
6
the most beneficial to the droplet rolling on the surface, which is the quite similar to
7
droplets on lotus leaves. As is shown in Figure 4a, c and d, when the first generation
8
structure is made of hydrophobic material (PDMS), whether the second generation
9
structures and the base are made of hydrophobic material (Si) or hydrophilic material
10
(SiO2), the C-C state is located at the lower left corner of the diagram. That is to say, when
11
b1 / a1 and b2 / a2 are designed to be small, the surfaces show superhydrophobicity
12
(ACAs >150°) and very low water roll-off angles (C-C state). As is shown in Figure 4b,
13
when the first generation structure is made of hydrophilic material (PC), although the
14
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surfaces are difficult to show superhydrophobicity and droplet-easy-rolling property
2
(ACAs
